Stochastic Formulation For Particle Kinetics Of Wall-bounded Granular Flow

The sediment transport near the bed is one of the substantial problems in the theory of sediment transport. The solid-fluid two phases flow is influenced by the boundary condition highly. The boundary bounds and suspends the sediment, near which the depletion and sedimentation occur. Many laboratory experiments and numerical simulations show that the particle motion near the wall is different from the one far from the wall. The wall boundary can induce an additional force on the particle with a drift velocity. In addition, the distributions of the position and velocity are no longer normal distribution as in the usual.Based on the stochastic process, a model of wall-bounded granular flow is built in the framework of random walk. Moreover, a Kramers equation induced from the skew Gaussian process is built to describe the probability density function of the position and velocity of the particle bounded by the wall. Three asymptotic solutions of such skew Kramers equation are given and the kinetic theory of the wall-bounded granular flow is developed.The motions of wall-bounded particles are depicted in the framework of random walk. A drift velocity deviating from the wall is induced due to the effect of the reflecting boundary,and the distributions of the particle position and concentration are skew, both of which are in agreement with the experiment observation。An approximate solution of the probability density function of the position of the wall-bounded particle is deduced. The structure of this solution indicates that the skew normal distribution is the suit to approximate the statistical feature of the particle near the wall. Therefore, an incremental skew Gaussian process is defined, based on which a stochastic differential equation is established to depict the particle motion near the wall.The Kramers equations describing the wall-bounded particle motion is drawn from the skew Gaussian stochastic process, where the probability density function in the phase space is taken as the variable. The kinetic theory of wall-bounded granular flow is established。Assuming the exterior force is potentialforce, three asymptotic solutions of the skew Kramers equations are deduced under the conditions of ignoring the collisional term, equilibrium state and steady flow respectively. Under the condition of ignoring the collision documents the effects of the particle collision, the initial and boundary condition of the Kramers equation do not change the formula of the velocity distribution, where the evolution of the interaction can't be considered when the particles with different velocity distribution collide. In the second condition as equilibrium state, the particle behavior adapts instantaneously to a local state, in which the particle is not affected by the environment dependently. Therefore, though an average drift velocity of the overall particles globally exists, the velocity distribution of individual particle is normal. Under the condition of steady flow, a general solution is taken considering the particle would tend to equilibrium state after a time of the order as relaxation time. So the series term with high order of relaxation time represents the effect of the wall, resulting in the skew velocity distribution deviating from the wall.